I recently looked into taking my 3-yr old MFT filly to a trainer and he insists that the horse be trained in a canter to "round" out all her gaits. Is it appropriate to train the canter on a gaited horse? I hear so many pros and cons.
There are no hard and fast rules regarding canter. As usual, training should depend on what suits the individual horse. What is your trainer experienced at doing? Training for the fox trot? General gaited? Or is he not specifically trained to work with gait at all? If it is the former, then take a 'wait and see' attitude--but make certain NO artificial gimmicks or devices are introduced into the training process. This means starting with a simple snaffle bit, light keg shoes, no weights, etc. A humane curb bit can be introduced after the horse has been taught to carry itself well under a rider.
If the trainer is not experienced with gait, then you need to understand where the canter may--or should not--fit into the picture. If a horse is quite pacey, then teaching it to canter can be very helpful as it trains the animal to break up the lateral set of legs. On the other hand, many more diagonally oriented horses (trottier) should have the gait well established BEFORE canter work is undertaken. This is because they tend to want to shortcut the very hard process of learning and establishing gait by picking up a canter lead instead. The result is sort of a 'hoppy-skippy' half canter, half gait MESS.
For that reason I honestly believe it's better for the owner to get 30 or 60 days of very basic walk, turn, stop training on their horse, with maybe slow/faster walk transitions thrown in, and then plan to do the rest of the training themselves. No one will care about your horse as much as you do!
The best early gait training consists of riding the (comfortably tacked) horse for many, many miles at a good active walk. As the horse becomes conditioned for the four-beat walk, ask for increasing speed and collection. Over time--usually several months'--if the horse is not permitted EVER to trot or pace (or even canter), then it will be developing all the skills and conditioning necessary for gaiting. As you ask for speed with collection, at some point the animal will simply start to gait. It's really THAT simple. The only 'trick' is being patient enough to allow the process enough time to be done correctly. No matter how strongly naturally gaited a horse is, this process should take place over several months' time to permit the horse to grow up into its own best natural gait, without danger of long-term breakdown or gait confusion.
I hope this was helpful. Many happy--and smooth--trails to you and your filly!